Karim Benzema equalised with the last kick of the match to book its place in the last eight and set a record of 40 games without defeat, surpassing the tally set by Luis Enrique's Barcelona.
Madrid was without Cristiano Ronaldo and Luka Modric as Zidane appeared to put faith in the 3-0 lead built up in the first leg, only to see his side fall behind nine minutes in through a Danilo own goal.
Sevilla attacked relentlessly and fully deserved its half-time advantage, although Wissam Ben Yedder in particular wasted glorious chances to build a bigger lead that could have made Madrid more nervous.
Marco Asensio levelled after the break, racing from his own half to slot home in a style reminiscent of Gareth Bale's final-winning strike from 2014, but debutant Stevan Jovetic volleyed Sevilla back in front moments later.
Jorge Sampaoli's side continued to press while Madrid wasted good openings on the counter-attack, and Vicente Iborra turned in with 13 minutes left to leave the visitor fearing a late collapse.
But a penalty from Sergio Ramos and a deflected shot after a fine run from Benzema ensured Madrid progresses with its remarkable unbeaten run intact.
Kiko Casilla, who kept his place in the side after the first leg, had to beat a Sergio Escudero effort away after only 30 seconds as Sevilla set about giving themselves hope of a shock comeback.
Joaquin Correa had the keeper at full stretch as the Sevilla pressure grew and Madrid soon handed them a lifeline. Danilo stretched to meet Pablo Sarabia's swerving cross from the right but contrived only to head into his own net.
Asensio brought a fine save from David Soria with a dipping shot from distance as Madrid began to settle, before Mariano - starting in place of Ronaldo - almost blocked a Soria kick into the back of the Sevilla net.
Casilla twice denied Ben Yedder in quick succession and Toni Kroos fired inches wide at the other end as the half continued at a frenetic pace.
Sevilla stepped up the pressure early in the second half but it was hit by a superb sucker punch from Asensio just three minutes after the restart. The 20-year-old collected Casilla's clearance from a corner, surged into Sevilla's half and away from Luciano Vietto before slotting with composure through the legs of Soria and into the bottom corner.
If the home fans inside the Ramon Sanchez Pizjuan were deflated, that feeling lasted only five minutes. Escudero found space down the left and curled a cross towards the unmarked Jovetic, who volleyed expertly past Casilla from six yards out.
Jovetic was denied a second by a low stop from Casilla before Ben Yedder glanced a promising header wide as Sevilla continued to believe in a miraculous comeback, although Alvaro Morata blazed over when played clean through by Danilo.
Iborra pounced on Casilla's fumbled effort to parry a low cross from the right to give the home side renewed hope but, after Casemiro was bundled over by Matias Kranevitter, Ramos chipped the ball confidently into the net before celebrating pointedly in front of his old supporters.
And with seconds remaining Benzema danced his way through the heart of the Sevilla defence and hit a deflected effort that wrong-footed Soria and sent the Madrid bench into raptures.
Sevilla was hugely disappointed at full-time but will have the chance to take revenge on Monday (AEDT), when LaLiga's top two face off at the same venue.